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### Identifying Algebraic Properties

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Date: 12/29/96 at 19:43:19
From: Kenneth L McConnell
Subject: Pre-Algebra Help

Dr. Math,

My daughter is in a pre-algebra class in the eighth grade.  I am
trying to understand the following problem so I can help her with a
homework assignment.  Any information you could give me regarding the
following would be greatly appreciated:

Choose which property is used to get each step of the solution:

d) Op-Op property
e) Property of Opposites

(-3 + 8) + -(-3)

=(-3 + 8) + 3    Answer__ Why?

= -3 + (8 + 3)   Answer__ Why?

= -3 + (3 + 8)   Answer__ Why?

= (-3 + 3) + 8   Answer__ Why?

= 0 + 8          Answer__ Why?

My primary request is for a definition of "a" through "e" and how they
apply to the above.

Thank you,

Ken McConnell
```

```
Date: 12/30/96 at 10:23:03
From: Doctor Rob
Subject: Re: Pre-Algebra Help

Glad that you are helping your daughter.  Perhaps you will both learn
something in the process!

Your daughter's textbook should be the source for the definitions of
the properties you list.  The first three properties are standard,
while the last two are not, so I can only guess about the meanings of
the last two.

a) Additive Identity of Zero (STANDARD)

Given any number, if you add zero to it, you get that number back.  In
equations, for any number a, a + 0 = 0 + a = a.

b) Associative Property of Addition (STANDARD)

Given any three numbers to be added, the result of adding the sum of
the first and second to the third is the same as adding the first to
the sum of the second and third.  In equations, for any three numbers
a, b, and c, (a + b) + c = a + (b + c).

c) Commutative Property of Addition (STANDARD)

Given any two numbers, the result of adding the first to the second
is the same as the result of adding the second to the first.  In
equations, for any two numbers a and b, a + b = b + a.

d) Op-Op property  (NON-STANDARD)

Given any number, the opposite of its opposite is the number itself.
In equations, for any number a, -(-a) = a.

e) Property of Opposites (NON-STANDARD)

Given any number, the sum of it and its opposite is zero.  In
equations, for any number a, a + (-a) = 0.

Now, given this, I think you can figure out which reason goes with
which operation.  Just match the patterns in the properties with the
patterns in the expressions, and see which ones fit.

By the way, underlying all of the above reasoning is the following
principle:  when two numbers are equal, any expression involving the
first will remain the same when the second is substituted for it.  For
example, in the first expression, you find -(-3), and from d) you know
that -(-3) = 3, so by this substitution principle, you can replace
-(-3) by 3 without changing the value of the expression.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra
Middle School Definitions

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